# Computers & Mathematics with Applications

Published by Elsevier

Online ISSN: 0898-1221

Published by Elsevier

Online ISSN: 0898-1221

Publications

Article

"In this paper we present a class of rapidly convergent numerical schemes to solve the Sharpe-Lotka model equation from age-dependent population dynamics. This work is based on spline approximation techniques described by Banks and Kappel for functional differential equations. We provide a motivation for and description of a generalized problem which under suitable conditions is equivalent to the Sharpe-Lotka problem, we describe approximation in general which exploits the Hilbert space structure in which the generalized problem is set, and for a particular space of approximating functions we obtain estimates on the rates of convergence."

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Article

Efficient flow of red blood cells (RBCs) and white blood cells (WBCs) through the microcirculation is necessary for oxygen and nutrient delivery as well as immune cell function. Because blood is a dense particulate suspension, consisting of 40% RBCs by volume, it is difficult to analyze the physical mechanisms by which individual blood cells contribute to the bulk flow properties of blood. Both experimental and computational approaches are hindered by these non-Newtonian properties, and predicting macroscopic blood flow characteristics such as viscosity has historically been an empirical process. In order to examine the effect of the individual cells on macroscopic blood rheology, we developed a lattice Boltzmann model that considers the blood as a suspension of particles in plasma, accounting explicitly for cell-cell and cell-wall interactions. Previous studies have concluded that the abundance of leukocyte rolling in postcapillary venules is due to interactions between red blood cells and leukocytes as they enter postcapillary expansions. Similar fluid dynamics may be involved in the initiation of rolling at branch points, a phenomenon linked to atherosclerosis. The lattice Boltzmann approach is used to analyze the interactions of red and white blood cells as they flow through vascular networks digitized from normal and tumor tissue. A major advantage of the lattice-Boltzmann method is the ability to simulate particulate flow dynamically and in any geometry. Using this approach, we can accurately determine RBC-WBC forces, particle trajectories, the pressure changes in each segment that accompany cellular traffic in the network, and the forces felt by the vessel wall at any location. In this technique, intravital imaging using vascular contrast agents produces the network outline that is fed to the lattice-Boltzmann model. This powerful and flexible model can be used to predict blood flow properties in any vessel geometry and with any blood composition.

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Article

The stochastic solution to a diffusion equations with polynomial coefficients is called a Pearson diffusion. If the first time derivative is replaced by a Caputo fractional derivative of order less than one, the stochastic solution is called a fractional Pearson diffusion. This paper develops an explicit formula for the covariance function of a fractional Pearson diffusion in steady state, in terms of Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long range dependent, with a correlation that falls off like a power law, whose exponent equals the order of the fractional derivative.

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Conference Paper

Terminal-pair reliability (TR) in an asynchronous transfer mode
(ATM) virtual path (VP) network corresponds to probabilistic
quantization of robustness between two VP terminators, given a VP layout
and failure probabilities of physical links. Existing TR algorithms are
not viable for ATM VP networks owing to either high complexity or
failure dependency among the VPs. The paper proposes two efficient
algorithms for TR computation between two VP terminators by means of
variants of path-based and cut-based partition methods which have been
effectively used for TR computation in traditional networks. The first
variant, called the path-based virtual path reliability (PVPR)
algorithm, partitions the search space based on a physical path
embedding the shortest route of VPs from the source to the destination
terminator. The second variant, called the cut-based virtual path
reliability (CVPR) algorithm, in lieu, performs the partition on the
basis of a physical cutset separating the source from the remaining
terminators. In both algorithms, each subproblem is recursively
processed by means of partition until the source and destination
terminators are contracted or disconnected. Experimental results reveal
that, CVPR outperforms PVPR with respect to computation time. Moreover,
compared to one of the most promising TR algorithms, both CVPR and PVPR
exhibit superior performance. The two algorithms and their promising
results consequently facilitate the real-time computation of the
reliability or robustness of ATM VP networks

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Conference Paper

Many semi-supervised learning algorithms only consider the distribution of words frequency, ignoring the semantic and syntactic information underlying the documents. In this paper, we present a new multi-view approach for semi-supervised document classification by incorporating both semantic and syntactic information. For this purpose, a co-training style algorithm Co-features is proposed, in active querying, we assign a weight to each sample according to its uncertainty factor, the most informative samples are selected and labeled by other "teachers". In contrast to batch training mode, an incremental Naive Bayes update method is realized, which allow more efficient classifier training even with large pool of unlabeled data. Experimental results show that our algorithm works successfully on Reuters-21578 and WebKB, and is superior to Co-testing in the learning efficiency.

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Conference Paper

An optimal parallel algorithm for computing all-pair shortest paths on doubly convex bipartite graphs is presented here. Our parallel algorithm runs in O(log n) time with O(n<sup>2</sup>/log n) processors on an EREW PRAM and is time-and-work-optimal. As a by-product, we show that the problem can be solved by a sequential algorithm in O(n<sup>2 </sup>) time optimally on any adjacency list or matrix representing a doubly convex bipartite graph. The result in this paper improves a recent work on the problem for bipartite permutation graphs, which are properly contained in doubly convex bipartite graphs

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Conference Paper

Defines a validity measure for fuzzy criterion clustering which is
a novel approach to fuzzy clustering that in addition to being
non-distance based addresses the cluster validity problem. The model is
then recast as a bilevel fuzzy criterion clustering problem. The authors
propose an algorithm for this model that solves both the validity and
clustering problems. The authors' approach is validated via some sample
problems

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Conference Paper

Protocol testing leads to the synchronization problem should test
sequences be applied to multiple distanced testers, namely under the
multi-party configuration. This paper presents a novel synchronization
paradigm which seamlessly unifies two synchronization techniques,
self-synchronizable sequences and external synchronization operations.
To demonstrate the viability of the proposed paradigm, we present the
generations of two synchronizable sequences: the synchronizable
preamble, and the synchronizable distinguishing sequences. The paper
shows that the complexities of the two sequences generations are
polynomial-bounded

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Article

In this short paper, a new Lagrangian function is reported which is particularly suited for large-scale nonconvex optimization problems with separable structure Our modification convexifies the standard Lagrangian function without destroying its separable structure so that the primal-dual decomposition technique can be applied even to nonconvex optimization problems. Furthermore, the proposed Lagrangian results in two levels of iterative optimization as compared with the three levels needed for techniques recently proposed for nonconvex primal-dual decomposition.

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Conference Paper

We consider a wide range of electromagnetic problems of scattering
by locally inhomogeneous bodies, whose dielectric and magnetic
properties are characterized by arbitrary distributed tensors
εˆ(x) and μˆ(x). To simulate the problems, we use
singular integral equations over the domain Q of nonhomogeneity. Both 2D
and 3D are examined in the same manner that includes an integral
formulation of the problem, investigation of the solvability and
uniqueness of the solution, an iterative numerical method. While 3D
problems are formulated by a general equation, different integral
equations can be considered for inhomogeneous 2D problems, depending on
properties of the media and source field polarization

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Conference Paper

A variant of Gaussian elimination (GE) method called the
successive Gaussian elimination (SGE) algorithm, for the parallel
solution of linear equations, is presented. Unlike the conventional GE
algorithm, the SGE algorithm does not have a separate back-substitution
phase-which requires O(N) steps using O(N) processors or O(log<sub>2
</sub><sup>2</sup>N) steps using O(N<sup>3</sup>) processors-for solving
a system of N linear algebraic equations. The SGE algorithm replaces the
back-substitution phase by only one step-division-and possesses
numerical stability through partial pivoting. Finally, an efficient
scheduling scheme for assigning the computational tasks in the SGE
algorithm on to the processors in a multiprocessor system is given

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Article

In this paper we generalize Bor’s result by using the correct definition of absolute summability.

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Article

A set theory called Real Set Theory is defined in which Generalized Continuum Hypothesis and Axiom of Choice hold good.

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Article

Numerical simulations of mathematical models can suggest that the models are chaotic. For example, one can compute an orbit and its associated finite-time Lyapunov exponents, and these computed exponents can be positive. It is not clear how far these suggestions can be trusted, because, as is well known, numerical methods can introduce spurious chaos or even suppress actual chaos. This focused review examines the fidelity of numerical methods. We look at the didactic example of the Gauss map from the theory of continued fractions, which allows a simple examination of backward error analysis for discrete dynamical systems and gives a clear picture of the effects of floating-point arithmetic. A similar use of backward error analysis, in the form of defect control, gives a useful understanding in the case of continuous dynamical systems. Finally, we discuss limitations of this ‘backward’ point of view.

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Article

An analytic study on linear systems of fractional differential equations with constant coefficients is presented. We briefly describe the issues of existence, uniqueness and stability of the solutions for two classes of linear fractional differential systems. This paper deals with systems of differential equations of fractional order, where the orders are equal to real number or rational numbers between zero and one. Exact solutions for initial value problems of linear fractional differential systems are analytically derived. Existence and uniqueness results are proved for two classes. The presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approaches.

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Article

This paper is concerned with the development of an algorithm for forecasting discrete stock and flow data generated by a higher order continuous time system from a sample of stock and flow data. The algorithm is shown to be optimal in the sense that the forecasts are exact maximum likelihood estimates of the conditional expectations of the post sample observations, conditional on all the information in the sample, when the innovations are Gaussian. It is also highly efficient computationally when used in conjunction with recently developed estimation methods.

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Article

We establish several types of a a priori error bounds for multiquadric and related interpolators. The results are stated and proven in the general multivariate case. These estimates show, for example, that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data. We also include numerical experiments which illustrate the theoretical results.

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Article

A mathematical puzzle from a recent issue of the New Scientist magazine is solved by combining the theory of permutations with Prolog's symbolic and other computational facilities. The scheme studied is interesting because it shows that the power of the generate-and-test approach, a rather crude approach known from Artificial Intelligence, is greatly enhanced if it is supplemented by some topical knowledge from the field of study. The puzzle involves searching for matrices with certain patterns, leading to the study of permutation types. The suggested route allows for the solution of a generalized version of the original puzzle.

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Article

This paper presents a constraint programming approach to the Enigma 1225, a mathematical puzzle published in the New Scientist magazine in February 2003. An approach based on Prolog was published recently. In this paper we give a constraint programming perspective on the problem, highlighting the differences between the two methodologies. We show how problem-specific knowledge can be easily incorporated into a constraint-based approach, giving an efficient constraint model for the generalized version of the puzzle. From the constraint programming point of view, the Enigma 1225 puzzle exhibits interesting symmetries, that can be eliminated using only a small number of constraints added to the model. Furthermore, properties of the puzzle can be used to derive a strong constraint propagation scheme that limits the search once an optimal solution has been found.

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Article

It is well known that the linear extension majority relation of a partially ordered set (P,≤P) can contain cycles when at least 9 elements are present in P. Computer experiments have uncovered all posets with 9 elements containing such cycles and limited frequency estimates for linear extension majority cycles (or LEM cycles) in posets on up to 12 elements are available. In this contribution, we present an efficient approach which allows us to count and store all posets containing LEM cycles on up to 13 elements.

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Article

An n job, single machine scheduling problem in which each job has a distinct due date, di, is studied in this paper. The objective is to determine an optimal schedule π0s for a set of jobs, S, such that the total absolute deviation of the schedule is minimized. This objective function is based on the due date value and on the earliness or tardiness of each job in the selected sequence. This paper presents a bounding scheme for the calculation of different lower bounds based on the overlap elimination procedure on a Just-In-Time schedule. Properties and theorems of the overlap elimination procedure are also provided. Finally, a numerical example is illustrated and some extensions of the approach are also discussed.

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Article

The Adaptive Neural Fuzzy Inference System (ANFIS) is used to design two vague systems, namely thermal comfort and group technologies in production and operations management. Results show that both systems can be modeled successfully by the combined use of a fuzzy approach and neural network learning.

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Article

In this paper, 14-velocity and 18-velocity multiple-relaxation-time (MRT)
lattice Boltzmann (LB) models are proposed for three-dimensional incompressible
flows. These two models are constructed based on the incompressible LBGK model
proposed by He et al. (Chin. Phys., 2004, 13: 40-46) and the MRT LB model
proposed by d'Humi\`{e}res et al. (Philos. Trans. R. Soc., A, 2002, 360:
437-451), which have advantages in the computational efficiency and stability,
respectively. Through the Chapman-Enskog analysis, the models can recover to
three-dimensional incompressible Navier-Stokes equations in the low Mach number
limit. To verify the present models, the steady Poiseuille flow, unsteady
pulsatile flow and lid-driven cavity flow in three dimensions are simulated.
The simulation results agree well with the analytical solutions or the existing
numerical results. Moreover, it is found that the present models show higher
accuracy than d'Humi\`{e}res et al. model and better stability than He et al.
model.

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Article

A family of deterministic algorithms is introduced, designed to solve the global optimisation problem for Lipschitz continuous functions of many variables. All the algorithms can be considered as generalisations of the bisection method: they proceed via a sequence of brackets whose infinite intersection is the set of global optima. Brackets are unions of similar simplexes. Acceleration methods, convergence properties and optimality questions are considered.

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Article

We have designed and implemented a set of highly efficient and highly scalable algorithms for an unstructured computational package, the PSAS data simulation package, as demonstrated by detailed performance analysis of systematic runs up to 512 nodes of an Intel Paragon. The preconditioned Conjugate Gradient solver achieves a sustained 18 Gflops performance. Consequently, we achieve an unprecedented 100-fold reduction in time to solution on the Intel Paragon over a single head of a Cray C90. This not only exceeds the daily performance requirement of the Data Assimilation Office at NASA's Goddard Space Flight Center, but also makes it possible to explore much larger and challenging data assimilation problems which are unthinkable on a traditional computer platform such as the Cray C90.

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Article

Computer processing of large non-preedited natural language texts has often been limited either to managing and editing or to analysing basic levels of content (indexes, concordances, clusters, etc.). Few systems approach syntactic information, even less semantic information. Because of the complexity and the originality of the underlying semantic information of any text it is not possible to import directly the A.I. and computational semantic concepts. It is necessary to explore news paths. The research presented here is oriented toward the understanding of certain semantic aspects in computer text processing (words and meaning representation and inference patterns). This is done through a model theoretic approach embedded in an algebraic language. The hypothesis which governs the concepts and the distinctions is the following: discourse in a text constitutes a semantic space built of an ordered set of sentences which are of different logical types and which present a specific pattern of coherence expressible in a syntactic manner.

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Article

A number of stationary localized solutions to the well-known pattern-forming gradient system mentioned in the title have been found. Their search is based on the theory of homoclinic orbits to a saddle-focus equilibrium and some results of linear symmetric differential operators with decaying coefficients along with computer simulations.

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Article

A new type of network flow theory is proposed, where no cancellation of flows in an edge is admitted when two or more flows are superposed. It offers a general framework in which to discuss congestion, blocking flows, etc., in a network. We will call the flows in this framework “uncontrollable flows” because they possess some basic properties of the flows which selfish and stubborn users, or users in emergency situations, generate in a network.The primary aim of this paper is not to develop “mathematics” but to introduce a new viewpoint from which to give another look at network flow problems. However, a number of interesting and challenging mathematical problems naturally arise in so doing.What kind of practical problems will be the dual of the concept of uncontrollable flows is also briefly discussed. © 1900 Elsevier Science Ltd. All rights reserved.

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Article

Adaptive and fault-tolerant schemes for routing messages in a 2D torus interconnection network for distributed memory multi-computers (message passing concurrent computers) are presented. For the adaptive scheme, two new techniques, channel switching and dimension switching, are developed and proved deadlock-free. For the fault-tolerant scheme, a message can be rerouted to a virtual destination, which in turn sends the message to the real destination. This scheme can tolerate all single faults and many multiple faults, and is deadlock-free. The two routing schemes are suitable for the high performance virtual cut-through and wormhole routing. The required hardware overhead for realizing the fault-tolerant scheme is small and no time penalty is paid in the fault-free case.

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Article

Fourth-order compact finite difference schemes are employed with multigrid techniques to simulate the two-dimensional square driven cavity flow with small to large Reynolds numbers. The governing Navier-Stokes equation is linearized in streamfunction and vorticity formulation. The fourth-order compact approximation schemes are coupled with fourth-order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods.

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Article

Phase change problems are of practical importance and can be found in a wide range of engineering applications. In the present paper, two proposed numerical algorithms are developed; the first one is general for phase change problems, while the second one is for ablation problems. The boundary elements method is used as a mathematical tool in conjunction with the proposed algorithms. Two test examples were solved and the results agree with the physics of the problems.

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Article

Growth conditions are imposed on f such that the following boundary value problem: (−1)my(2m) = f(t, y), αi+1y(2i)(0) − βi+1y(2i+1)(0) = γi+1y(2i)(1) + δi+1y(2i+1)(1) = 0, 0 ≤ i ≤ m−1, has an arbitrary number of positive solutions.

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Article

Block Runge-Kutta methods with minimal phase-lag for first order periodic initialvalue problems are developed. It should be noted that the new methods are based on the Runge-Kutta methods of algebraic order three, and on a new error estimate introduced in this paper. The numerical results indicate that these new methods are efficient for the numerical solution of differential equations with periodic solutions, using variable step size.

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Article

A generalization of Ostrowski's inequality for Lipschitzian mappings and applications in numerical analysis and for Euler's Beta function are given.

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Article

Multibody systems are considered which involve combinations of rigid and elastic bodies. Discretizations of the PDEs, describing the elastic members, lead to a semidiscrete system of ODEs or DAEs. Asymptotic methods are introduced which provide a theoretical basis of various known engineering results for the ODE case. These results are then extended to the DAE case by means of suitable local ODE representations. The recently developed MANPAK algorithms for computations on implicitly defined manifolds form the basic tools for a computational method which provides consistent approximate solutions of the semidiscrete DAE that satisfy all constraints and are close to the smooth motion and an average solution. Several numerical examples indicate the effectiveness of this asymptotic method for elastic multibody systems.

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Article

We propose and analyze an overlapping Schwarz preconditioner for the $p$ and
$hp$ boundary element method for the hypersingular integral equation in 3D. We
consider surface triangulations consisting of triangles.We prove a bound for
the condition number that is independent of the mesh size $h$ and polynomial
order $p$. Additionally, we provide an extension to adaptive meshes based on a
local multilevel preconditioner for the lowest order space. Numerical
experiments on different geometries support our theoretical findings.

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Article

A formal language approach for representing three-dimensional (3D) curves is presented. Based on the chain code for representing 3D curves defined in [1], we propose an approach for mapping 3D curves into strings. This mapping allows us to have a unique curve descriptor, which is invariant under translation and rotation. Also, it is possible to use inverse and mirroring operators and the use of formal language techniques for 3D curve generation and analysis. Finally, we present a result of this approach to represent and to generate polygonal sequences convergent to cube-filling Hilbert curves.

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Article

A measure of compactness for 3D (three dimensional) shapes composed of voxels, is presented. The work proposed here improves and extends to the measure of discrete compactness [1] from 2D (two dimensional) domain to 3D. The measure of discrete compactness proposed here corresponds to the sum of the contact surface areas of the face-connected voxels of 3D shapes. A relation between the area of the surface enclosing the volume and the contact surface area, is presented. The concept of contact surfaces is extended to 3D shapes composed of different polyhedrons, which divide space generating different 3D lattices. The measure proposed here of discrete compactness is invariant under translation, rotation, and scaling. In this work, the term of compactness does not refer to point-set topology, but is related to intrinsic properties of objects. Finally, in order to prove our measure of compactness, we calculate the measures of discrete compactness of different volcanos (which are compared with their classical measures) from the valley of México using Digital Elevation Model (DEM) data.

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Article

A multigrid algorithm is presented for cell-centered discretizations of interface problems. Instead of constructing the coarse grid operators by means of the Galerkin approximation, the coarse grid operators are obtained by discretization on the coarse grids. The advantage of this approach is that we obtain M-matrices on all grids, and that the sparsity pattern of the fine grid matrix is retained on all grids. Moreover, the coarse grid operators are very easy to construct. Numerical results of several test problems are presented.

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Article

This paper presents a dynamic domain decomposition (D3) technique for implementing the parallelization of the piecewise parabolic method (PPM) for solving the ideal magnetohydrodynamics (MHD) equations. The key point of D3 is distributing the work dynamically among processes during the execution of the PPM algorithm. This parallel code utilizes D3 with a message passing interface (MPI) in order to permit efficient implementation on clusters of distributed memory machines and may also simultaneously exploit threading for multiprocessing shared address space architectures. 3D global MHD simulation results for the Earth’s magnetosphere on the massively parallel supercomputers Deepcomp 1800 and 6800 demonstrate the scalability and efficiency of our parallelization strategy.

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Article

An effective hp-adaptive finite-element (FE) approach is presented for a reliable and accurate solution of 3D electromagnetic scattering problems. The far field is approximated with the infinite-element method. This allows one to reduce the external domain (discretised with finite elements) to a minimum preserving the possibility of arbitrary reduction of the error as the method does not introduce modelling error. The work is focused on scattering from cavity backed apertures recessed in a ground plane. Near optimal discretisations that can effectively resolve local rapid variations in the scattered field can be obtained adaptively by local mesh refinements (so called h-type refinements) blended with graded polynomial enrichments (p-enrichments). The discretisation error can be controlled by a self-adaptive process, which is driven by a posteriori error estimates in terms of the energy norm or in a quantity of interest. The radar cross section (RCS) is an example of the latter. h- and p-adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for generic cavities are given and compared to previously published results.

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Article

In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1) (1999) 204–225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.

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Article

We report on experimental studies of the collision process between an incident bead and a three-dimensional packing of mono-size beads. The understanding of such a process and the resulting ejection of grains is, in particular, crucial to describe aeolian sand transport. We present here recent experimental results on the collision and ejection process when varying the angle and the speed of the incident bead. We performed numerical simulations of one bead collisions on the surface of a static granular medium. The simulations have been done for two and three dimensional packings of beads. The effects of the incident bead velocity, the shot angle, the mechanical parameters and the packing structure are analyzed for ordered and disordered 2D packings but only disordered 3D packings. The 2D results are in good agreement with available experimental data. The 3D simulations give good preliminary results about the shock wave propagation through the stacking and provides new insights in the ejection process (“splash function”).

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Article

We consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as where is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical 3). Here r is a remainder of order where is the anisotropic Rossby number (H0—height, L0—horizontal length scale, Щ0—angular velocity of background rotation, U0—horizontal velocity scale); where is the aspect ratio and is a Rossby number based on the horizontal length scale L0. The vector field which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases , τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl and velocity ; the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for and 2D turbulence for vertically averaged fields if the anisotropic Rossby number Roa is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T∗ of regular solutions to 3D Euler equations (T∗ → +∞, as Roa → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case.

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Article

By examining the irreducibility of a certain recurrence, we show that the hypergeometric function of the title cannot be represented by gamma functions.

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Article

Algorithms for computing peaks of certain statistics related to the 3x+1 problem are described, along with data on such peaks up to 56 trillion (5.6×1013). The data result from several years of computation. The design of the algorithms used illustrates several techniques for program optimization.

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Article

A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step.

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Article

Godunov-type methods and other shock capturing schemes can display pathological behavior in certain flow situations. This paper discusses the numerical anomaly associated to slowly moving shocks. We present a series of numerical experiments that illustrate the formation and propagation of this pathology, and allows us to establish some conclusions and question some previous conjectures for the source of the numerical noise. A simple diagnosis on an explicit Steger-Warming scheme shows that some intermediate states in the first time steps deviate from the true direction and contaminate the flow structure. A remedy is presented in the form of a new flux split method with an entropy intermediate state that dissipates the oscillations to a numerically acceptable level, and fix or reduce a variety of numerical pathologies.

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Article

In this paper, we focus our discussion on the parameterization reduction of soft sets and its applications. First we point out that the results of soft set reductions offered in [1] are incorrect. We also observe that the algorithms used to first compute the reduct-soft-set and then to compute the choice value to select the optimal objects for the decision problems in [1] are not reasonable and we illustrate this with an example. Finally, we propose a reasonable definition of parameterization reduction of soft sets and compare it with the concept of attributes reduction in rough sets theory. By using this new definition of parameterization reduction, we improve the application of a soft set in a decision making problem found in [1].

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Article

We present an efficient computational framework to quantify the impact of
individual observations in four dimensional variational data assimilation. The
proposed methodology uses first and second order adjoint sensitivity analysis,
together with matrix-free algorithms to obtain low-rank approximations of ob-
servation impact matrix. We illustrate the application of this methodology to
important applications such as data pruning and the identification of faulty
sensors for a two dimensional shallow water test system.

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